Let and be the coordinates of the mth pixel on the boundary of
a given 2D shape containing pixels, a complex number can be formed as
, and the Fourier Descriptor (FD) of this shape is
defined as the DFT of :
The top part of the following image shows the (horizontal, red-dashed curve) and (vertical, blue-continuous curve) coordinates of the pixels on the boundary of the shape, while the bottom part shows the real and imaginary parts of the frequency components, the Fourier descriptors (FD), of the boundary:
The following image shows the reconstruction of Gumby based on the first low frequency components (excluding the DC). Top: , and ; middle: , and ; bottom: , and .
It can be seen that the reconstructed figures using a small percentage of the frequency components are very similar to the actual figure, which can be reconstructed using one hundred percent of the FDs.
Fourier discriptor has the following properties:
If the 2D shape is translated by a distance :
If the 2D shape is scaled (with respect to origin) by a factor :
If the 2D shape is rotated about the origin by an angle :
If the starting point on the boundary is shifted from 0 to :